Category $\mcal O$ for polynomial toroidal algebras and its subalgebras

Abstract: In this paper we study Category $\mcal O$ for the polynomial toroidal Lie algebras and its $S,H$ type subalgebras. We classify irreducible objects of category $\mcal O$ as unique irreducble quotient of standard modules. Surprisingly, costandard objects of category $\mcal O$ arrises from Shen-Larsson type modules. We determine necessary sufficient conditions for irreducibility of Shen-Larsson modules. Finally appeling structure of Shen-Larsson modules and Soergel Tilting module theory of \cite{Soe}, we compute charcter formulas for irreducible modules and indecomposable Tilting modules of category $\mcal O$.

• The paper introduces category O for polynomial toroidal algebras by constructing standard, costandard, and tilting modules.
• It leverages triangular decompositions and explicit module realizations to derive detailed character formulas and analyze reducibility.
• The work extends classical BGG theory to polynomial settings, paving the way for further research in infinite-dimensional Lie algebras.

Category 𝒪 for Polynomial Toroidal Algebras and Its Subalgebras: A Technical Summary

Introduction and Theoretical Context

This work conducts a systematic investigation of the Bernstein-Gelfand-Gelfand (BGG) Category $\mathcal{O}$ for polynomial toroidal Lie algebras and their prominent subalgebras of special ($S$-type) and Hamiltonian ($H$-type) vector fields. While the representation theory of toroidal Lie algebras over Laurent polynomials and their full toroidal counterparts has seen notable development, there has been a substantial gap regarding polynomial analogs. The present paper addresses this by constructing and analyzing the representation theory, internal module structure, and character theory over the Lie algebras $L(\mathfrak{g},X_n)$, where $X_n$ ranges over $\{W_n, S_n, H_n\}$. These are semidirect sums involving a finite-dimensional simple Lie algebra $\mathfrak{g}$, the polynomial algebra $A_n$, central extensions, and subalgebras of the Witt algebra.

Category $\mathcal{O}$, central to the representation theory of semisimple Lie algebras, is extended in this analysis to $L(\mathfrak{g},X_n)$, leveraging the natural grading and triangular decompositions possessed by these algebras.

Structure of $S$0 and Its Category $S$1

A defining feature of $S$2 is its $S$3-grading inherited from $S$4. Its degree zero component $S$5 decomposes as a direct sum of $S$6, a reductive algebra $S$7, and the center generated by the $S$8. Particularly, $S$9 is explicit:

• For $H$0, it is $H$1,
• For $H$2, it is $H$3,
• For $H$4 (with $H$5 even), it is $H$6.

Category $H$7 for $H$8 consists of $H$9-graded modules with finite-dimensional graded pieces, local finiteness over the parabolic subalgebra $L(\mathfrak{g},X_n)$0, and weight space decomposition with respect to a maximal toral subalgebra.

Irreducible objects in this category are classified as unique irreducible quotients of standard modules (Verma-type modules), paralleling the classical BGG setting.

Construction of Standard, Costandard, and Shen-Larsson Modules

Standard modules $L(\mathfrak{g},X_n)$1 are induced from finite-dimensional irreducibles over the degree zero and negative-degree components, parameterized by dominant integral weights $L(\mathfrak{g},X_n)$2 for $L(\mathfrak{g},X_n)$3 and central characters $L(\mathfrak{g},X_n)$4. Each standard module admits a unique irreducible quotient, denoted $L(\mathfrak{g},X_n)$5, and these exhaust the simple objects in category $L(\mathfrak{g},X_n)$6.

Costandard objects are realized via contragredient duality and, interestingly, are shown to be naturally isomorphic to certain explicitly constructed Shen-Larsson type modules on $L(\mathfrak{g},X_n)$7, for finite-dimensional irreducible $L(\mathfrak{g},X_n)$8 over $L(\mathfrak{g},X_n)$9. The module action is a genuine lift of the classical Shen-Larsson construction from the theory of Cartan type Lie algebras to the semidirect polynomial setting. Exceptional modules, arising from specific singular highest weights (direct sums of fundamental weights), are shown to be reducible—a notable structural point.

A categorical framework encapsulating modules carrying compatible $X_n$0 and $X_n$1 actions is developed, facilitating analysis of reducibility and submodule structure.

Complete Classification and Structural Properties

A categorical study utilizing techniques from Skryabin, as well as analysis of the action of subalgebras analogous to $X_n$2, $X_n$3, and $X_n$4, yields a full classification:

• Shen-Larsson modules are irreducible unless their underlying $X_n$5-module corresponds to an exceptional weight.
• Exceptional modules decompose explicitly, with the de Rham sequence realized on $X_n$6-modules $X_n$7. These sequences are exact, and composition series and multiplicities are computed, generalizing classical results in the Cartan type case.
• The structure theorems for $X_n$8 and $X_n$9 analogs, including explicit description of composition factors for exceptional cases, align with the corresponding theory for differential operators and Hamiltonian Lie algebras.

Character Theory for Irreducible and Tilting Modules

A full character formula is derived for irreducibles in $\{W_n, S_n, H_n\}$0. For non-exceptional parameters, the formal character of an irreducible is a product of the character of the polynomial algebra (the "Fock space" term), the character of the finite-dimensional inducing module, and a Poincaré polynomial determined by the grading: $\{W_n, S_n, H_n\}$1 where $\{W_n, S_n, H_n\}$2 is the character of $\{W_n, S_n, H_n\}$3 with respect to the $\{W_n, S_n, H_n\}$4 action.

In the exceptional (reducible) cases, explicit alternating sum formulas for the characters of composition factors are given, determined recursively, e.g., for $\{W_n, S_n, H_n\}$5: $\{W_n, S_n, H_n\}$6 and similarly for $\{W_n, S_n, H_n\}$7 and $\{W_n, S_n, H_n\}$8, with more intricate multiplicities in the $\{W_n, S_n, H_n\}$9 case (reflecting the symplectic structure).

Tilting modules, constructed axiomatically via Soergel’s methods, are shown to exist uniquely for each standard module. Their character multiplicities with respect to standard modules translate via duality into multiplicities of composition factors in the costandard module under highest weight duality and the action of the longest Weyl element. The necessary semi-infinite character is constructed and shown to be unique (trivial for $\mathfrak{g}$0, $\mathfrak{g}$1; canonical for $\mathfrak{g}$2). Explicit formulas for tilting module characters are presented, and the multiplicities and linkage for the exceptional weights are fully derived.

Implications and Future Directions

This work significantly advances the representation theory of infinite-dimensional Lie algebras of mixed polynomial type by rigorously generalizing the theory of category $\mathfrak{g}$3 from finite and toroidal Lie algebras to the setting of polynomial-coefficient analogues. It makes several technically nontrivial contributions:

• The first complete classification of irreducibles and tiltings in category $\mathfrak{g}$4 for $\mathfrak{g}$5,
• Construction and analysis of costandard objects via explicit module realizations,
• Discovery of direct connections between costandard objects and Shen-Larsson modules,
• Exhaustive treatment of exceptional modules, including their precise structure and character.

The theoretical framework established here opens the way for further research into extension groups, block decomposition, and endomorphism algebras in these categories, as well as connections with related objects such as extended affine Lie algebras and their quantizations. The techniques for deriving character formulas may generalize to algebras with more intricate central extensions or to settings involving superalgebras, as indicated by the module category definitions and references.

Conclusion

This paper gives a definitive and detailed account of category $\mathfrak{g}$6 for polynomial toroidal Lie algebras $\mathfrak{g}$7 and their key subalgebras. The explicit module constructions, complete irreducible classification, and derivation of character formulas establish a foundational structure theory for these infinite-dimensional Lie algebras compatible with their homogeneous gradings and underpinned by methods from the classical and modular representation theory. The results both consolidate the theoretical landscape for polynomial vector field algebras and provide a platform for further advanced investigations into either their representation theory or related algebraic and geometric structures.

Reference: "Category $\mathfrak{g}$8 for polynomial toroidal algebras and its subalgebras" (2604.10764)